On Thursday March 10, Peter Cholak gave a beautiful talk at the Logic Seminar at the University of Michigan, on Rado’s path decomposition theorem and its effective content. I want to review here some of the results covered by Peter. Slides for another version of the talk can be found in Peter’s page. This is joint work by Peter, Greg Igusa, Ludovic Patey, and Mariya Soskova.

As usual, given a set , let denote the collection of 2-sized subsets of . If is a positive integer, an –coloring of (or simply, a coloring, if is understood) is a map (where we use ordinal notation, so ). We can think of this as a coloring using colors of the edges of the complete graph whose underlying set of vertices is . When , we have an even simpler interpretation: A 2-coloring is just a graph on .

Given an -coloring of , a path of color is a sequence of distinct elements of (which may be finite or infinite, or even empty, or of length 1) such that for all , if is defined, then . Note that this is a much weaker requirement than asking that be monochromatic (which would mean that for all ). Also, in what follows is either a finite number or . However, we do not require that the elements in the sequence be listed in their natural order: We may very well have that for some .

The starting point is the following observation:

Fact (Erdős).If is finite and is a 2-coloring of , then there are paths of color 0 and of color 1 such that every (vertex) appears in exactly one of the

In general, if is an -coloring of , we say that , , is a path decomposition of iff each is a path of color and every vertex appears in exactly one of the . Using this notion, what the fact states is that for any finite , any 2-coloring of admits a path decomposition.

Proof. Suppose the result holds for and is a 2-coloring of . We can then find paths and of color 0 and 1 respectively such that each appears in exactly one of the . We want to show that the same holds for the full coloring (which includes edges one of whose vertices is ) at the possible expense of having to modify the partial paths we currently have. If one of the is empty, this is clear. Assume then that and . The result is also clear if or . Finally, if and , consider . If this color is 0, we can let the paths be and . Similarly, if , we can let the paths be and . (This is perhaps most obvious if a picture is drawn.)

Rado’s paper is a generalization of this result and its countable version. The reference is

The paper opens indicating that Erdős sketched his proof to Rado; there does not seem to be an actual reference for Erdős’s proof. Rado proceeds to prove a more general version. I will only discuss here a particular case.

First, it should be noted that, unlike typical results in Ramsey theory where, once the case of two colors is handled, the argument easily generalizes to any number of colors, the proof above does not lift to more than two. The usual way of doing this lifting is by identifying all but one of the colors. This would result in two paths and , where along we only see color 0 and along we only see the other colors, but not 0. Let be the given coloring and be the set of vertices appearing in . If the restriction of to does not use color 0 we could indeed proceed inductively. But there is nothing to prevent 0 from being present as well, so the “easy” lifting argument actually breaks down.

The situation is indeed worse:

Theorem (Pokrovskiy).For any and any there is an and an -coloring of that does not admit a path decomposition.

Theorem (Rado).For any finite , any -coloring of admits a path decomposition.

As already mentioned, Rado’s result is more general, in particular allowing the use of countably many colors. However, the arguments that follow only apply directly to the stated version.

Before sketching the proof, note that even for , the result does not follow as usual from the finite version: Given a 2-coloring of , the standard approach would consist of letting be the paths resulting from successively applying Erdős’s theorem to the restrictions of to . But the inductive argument we presented allows the paths to be modified from one value of to the next, which means that we cannot ensure that the process will successfully identify (via initial segments) paths for the full coloring (the partial paths do not “stabilize”). Together with Pokrovskiy’s negative result just indicated, this leaves us with a curious Ramsey-theoretic statement to which the usual compactness arguments do not apply. (Its finite counterpart, Erdős’s result, is weaker in the sense that it only applies to two colors, and requires a different argument.)

Proof. Consider a nonprincipal ultrafilter on . The ultrafilter provides us with a notion of largeness. Given and a coloring , define for and the set of neighbors of in color as

Note that for any the partition and therefore there is exactly one such that is large (that is, it is in $\mathcal U$). For , define

,

and note that the partition .

We proceed by stages to define the paths as required. We set for all . In general, at the beginning of any given stage we have defined (finite) partial approximations to each path , say has length , with , using the convention that indicates that . For each , we will ensure that end extends (for all ), and simply set as the resulting path. Inductively, we require that each is a path of color , and that if , then .

Now, at stage , we simply consider the least not yet in any of the . There is a unique with . We set for all . If , then set . Finally, if , the point is that since and are both large, then so is their intersection (all we really need is that the intersection of sets in is nonempty). Let be a point in their intersection, and set . The induction hypothesis is preserved, and this completes stage of the construction.

It should be immediate that the so constructed indeed provide a path decomposition of , and this completes the proof.

It is interesting to note that the notation just developed allows us to give a quick proof of Ramsey’s theorem for pairs: Given a coloring , use notation as above, and note that for exactly one , the set is in . We argue that there is an infinite subset that is homogeneous for with color , that is, . Indeed, we can simply set , where the are defined recursively so that and for all .

As Peter indicated in his talk, these pretty arguments are somewhat dissatisfying in that invoking a nonprincipal ultrafilter is too strong a tool for the task at hand. He then proceeded to indicate how we can in fact do better, computationally speaking. For instance, if the coloring is computable, then we can find a path decomposition below . The key to this improvement comes from two observations.

First, we do not really need an ultrafilter to carry out the argument. It suffices to consider a set that is cohesive with respect to all the , meaning that is infinite and, for any , either or , where is the eventual containment relation: iff there is a finite subset of such that , in which case we say that is almost contained in .

The point is that we can replace all instances where we required that a set is in by the new largeness condition stating that is almost contained in . For instance, note that if are large, then so is their intersection. As before, for any there is a unique with large, and we can redefine as the set of such that

or, equivalently,

With these modifications, it is straightforward to verify that the proof above goes through. This shows that a path decomposition of is .

In more detail: Note first that these two conditions are indeed equivalent, and second, clearly the are pairwise disjoint since is infinite and, moreover, for all there is a unique such that :

Suppose that holds, and let be such that . Since is infinite, we can indeed find elements of larger than , and any such witnesses .

Conversely, if holds, then , because is cohesive and has infinite intersection with . But then holds, as wanted.

To see that any is in a unique , fix and use that is cohesive to conclude that if for all , then , which contradicts the infinitude of . It follows that for some and, since the are pairwise disjoint, this is unique. This proves that holds and therefore . Uniqueness follows from this same observation: If , then (as shown above) . But there is only one for which this is true.

The second observation is that there is an easy recursive construction of a set that is cohesive with respect to all the : Consider first . One of these sets is infinite (since their union is ), say , and let be its first element. Consider now

.

Their union is , so one of these sets is infinite, say . Let be its first element above . Etc. The set so constructed is as wanted. Note that this construction explicitly obtains an infinite set that, for each , is almost contained in one the , which is superficially stronger than being cohesive. However, as verified above, any set cohesive for all the must actually have this property.

Computationally, the advantage of this construction is that it makes explicit that all we need to access a cohesive set is an oracle deciding of any whether it is infinite. For computable , these are all questions.

Peter further refined this analysis in his talk via the notion of a set being over : This is any set such that for any uniformly computable sequence of pairs of sentences for such that at least one is true, there is an that predicts the true sentence of each pair in the sense that for all , if , then is true. In symbols, say that . The point of the notion is that a result of Jockusch and Stephen gives us that if then there is a cohesive set such that . The relevant paper is:

This shows that a path decomposition for a computable coloring can actually be found below (and more).

Peter concluded his talk by indicating how for special colorings the complexity can be further improved. For instance, say that a coloring is stable iff exists for all . One can check that for stable , we can use cofinite as a notion of largeness in the preceding arguments, and that a path decomposition can accordingly be found when is computable below . On the other hand, this is optimal, in that one can find a stable computable such that any path decomposition computes .

My student Summer Kisner completed her M.S. this term, and graduated on Saturday.

2013–5-18 Summer

Here is a copy of the slides she used on her defense. (The slides display incorrectly on my computer, but it seems to be a problem on my end. If it is not, please let me know, and I’ll see what I can do.)

Her thesis, Schur’s theorem and related topics in Ramsey theory, discusses Schur’s theorem, one of the first result in what we now call Ramsey theory. The result states that if the positive integers are partitioned into finitely many sets, , then for some , , there are integers (not necessarily different), all of them in , such that . One usually describes this in terms of colors: We color the positive integers with finitely many colors, and there is a monochromatic triple with .

This result is a cornerstone of Ramsey theory. It was significantly generalized by Rado (using the notion of partition regularity), and is connected to van der Waerden’s and Szemerédi’s famous results.

Nowadays, Schur’s theorem is typically proved as a corollary of Ramsey’s theorem. This is usually stated in terms of graphs, but I will use the notation from the partition calculus. Let denote the collection of -sized subsets of the set . Suppose that is infinite, and consider a coloring , where the set of colors is finite. Ramsey’s theorem asserts that under these assumptions, there is an infinite subset of that is homogeneous or monochromatic for , in the sense that assigns the same color to all -sized subsets of . In fact, we have finitary versions of this result: For any and any , if we only require that has size at least , then there is an such that it suffices to assume that has size at least . Even for , the study of Ramsey numbers, the least seen as a function of , proves to be incredibly difficult and computationally unfeasible. For example, if , then we know that , but its exact value is not known.

To deduce Schur’s theorem from Ramsey’s, let be such that, for any for coloring of using colors, there is a monochromatic set of size . The least such we denote . We want to show that if , then there is a monochromatic solution to the equation . In fact, we claim that it suffices to consider rather than the whole set of positive integers. Indeed, given a partition , consider the coloring of where if , then the set $\{a,b\}$ has color , where . By definition of , we can find such that all have the same color. Now notice that , that is, are monochromatic for the original coloring of .

An easy inductive argument gives us that , so this gives the upper bound for the so-called -th Schur number . To see the upper bound , note that , and verify inductively that : Suppose that , and consider a coloring of with . Fix an element , and note that for some color there are elements such that has color . Let be the set of all these , that is, has color . Note that if for some we have that has color as well, then is monochromatic with color . Hence we may assume that the coloring, restricted to , only uses colors. We are now done, since .

Schur’s original proof predated Ramsey, and gives a slightly better bound than . Indeed, from his proof, we obtain that .

In terms of lower bounds, one can quickly check by induction that . Indeed, , since . Given a coloring of using colors and without monochromatic triples, we describe a coloring of using colors , again without monochromatic triples. This gives the result. To define , start by letting for . Now let for , and finally let for .

Slightly better bounds are known. For example, Anne Penfold Street shows that in

Schur proved his theorem in order to establish a result related to Fermat’s last theorem, namely that it cannot be established by a naive argument involving modular arithmetic: Suppose that are integers, and . Then, for any prime , the equality holds modulo and, if is large enough, then we also have that . Hence, to prove that Fermat’s equation admits no solutions , it would suffice to show that there are arbitrarily large primes such that must divide one of . What Schur proved is that this is not possible and, indeed, for any and all sufficiently large primes , there are nontrivial solutions to Fermat’s equation modulo . To see this, let and let be the subgroup of consisting of -th powers. Then is union of cosets of , say where the displayed sets are disjoint. Note that . Now color using colors, by letting have color iff . By choice of , we have a monochromatic Schur triple, that is, there are such that and for some . But then there are , all nonzero modulo , such that , , and , so is a nontrivial solution to Fermat’s equation modulo .

It is actually an interesting problem to try and determine the optimal size of as a function of . Fourier-analytic methods give here the best known bounds. Cornacchia proved in 1909 that , at least if itself is prime.

Let me close with an open problem: We could consider a multiplicative (rather than additive) version of Schur’s theorem: For any there is an such that if is colored using colors, then there is a monochromatic set with . Indeed, this follows as a simple corollary of Schur’s result: Just note that , since we could just color the powers of two and apply the additive version. what if we combine the two? It is still open whether in any finite coloring of there are integers such that all receive the same color. This was originally asked by Hindman.

There are several possible ways in which one can try to generalize Ramsey’s theorem to larger cardinalities. We will discuss some of these generalizations in upcoming lectures. For now, let’s highlight some obstacles.

Theorem 1 (-Kakutani) In fact,

Proof: Let Let be given by

Then, if are distinct, it is impossible that

Theorem 2 (Sierpi’nski) In fact,

Proof: With as above, let be given as follows: Let be a well-order of in order type Let be the lexicographic ordering on Set

Lemma 3 There is no -increasing or decreasing -sequence of elements of

Proof: Let be a counterexample. Let be least such that has size and let be such that if then To simplify notation, we will identify and For let be such that but By regularity of there is such that for many

But if and then iff so It follows that has size contradicting the minimality of

The lemma implies the result: If has size and is -homogeneous, then contradicts Lemma 3.

Now I want to present some significant strengthenings of the results above. The results from last lecture exploit the fact that a great deal of coding can be carried out with infinitely many coordinates. Perhaps surprisingly, strong anti-Ramsey results are possible, even if we restrict ourselves to colorings of pairs.

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